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Full text of "Treatise On Analysis Vol-Ii"

22 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA Let CQxe i be a family of mappings of E into F (resp. R) whose supports form a locally finite family. Then the sum ]T /a(x) is defined for all x e E, ae I because it contains at most finitely many non-zero terms. We denote by £ fa the function x\-+ £.£(#). If F is R or a normed space (or more generally ael ael a topological vector space (12.13)), and if each fa is continuous on E, then so is /= £ y^: for given x e E there exists a neighborhood V of x which meets only ael finitely many of the Supp(^), and hence there is a finite subset H of I such that/0>)=:£/a(7) for all jeV. aeH A continuous partition of unity on E is by definition a family (/^)a 61 of continuous mappings of E into [0, 1], such that the supports of the fa form a locally finite family, and such that £ fa(x) = 1 for all jc e E. If (Aa)ae t is an ael open covering of E indexed by the same set I, then the partition of unity C/a)« e i is said to be subordinate to the covering (Aa)a 6 j if we have Supp(^) c Aa for all a e I. (12.6.3) Let (An) Z>£ an at most denumerable locally finite open covering of a metrizable space E. Then there exists a continuous partition of unity (fn) on E subordinate to (An). Let (Brt) be an open covering of E, such that Bw cz Aw for all n (12.6.2). It is clear that the covering (Bn) is locally finite. By (4.5.2), for each n there exists a continuous mapping hn:E-» [0, 1] such that hn is equal to 1 on Bn and equal to 0 on E - Art. If we put gn = (hn — %)+, then Supp(#n) is contained in the set of points x such that hn(x) ^ i, and hence is contained in An. Let g = £ gn. n Since the sets Brt cover E, we have g(x) > 0 for all x e E, and therefore the functions^ = gjg are defined and continuous on E, and form a partition of unity with the required properties. (12.6.4) Let E be a metrizable space, K a compact subset ofE an a finite covering ofK by open sets ofE. Then there exist m continuous map- pings fk: E-*> [0, 1] such that Supp(j^) c: Ak for 1 <^ k g m, and such that m m %fk(x)^lforallxeEand £/*(*)= IforallxeK. Take a continuous partition of unity C4)ogfc^m subordinate to the open covering of E consisting of A0 = E — K and the Ak (1 ^ k g m).e Bk for some k ^ n. Hence the sets BM cover E.